Open Access

ARTICLE

Investigation on an Accelerated Scheme for Solving Time-Dependent Systems

Montri Maleewong¹, Sirod Sirisup²

1 Department of Mathematics, Faculty of Science Kasetsart University, Bangkok, Thailand
2 Large-Scale Simulation Research Laboratory National Electronics and Computer Technology Center, Thailand, Corresponing author: sirod.sirisup@nectec.or.th

Computer Modeling in Engineering & Sciences 2010, 65(2), 193-216. https://doi.org/10.3970/cmes.2010.065.193

Abstract

In this paper, we describe our investigation of an "on-line" POD-assisted projective integration method for solving a nonlinear PDE. Using the on-line method, we have computed the representative POD modes without assuming knowledge of the underlying slow manifold along the integration process. This approach is based on the "equation-free" framework where the governing PDE does not need to be projected onto the POD bases in order to build a reduced-order model. The main objectives of this study were to investigate the effectiveness of the method in reducing the computational time required for numerically solving a nonlinear PDE. Here, the one-dimensional viscous Burgers' equation is chosen as the time-dependent illustrative prototype. The numerical results from this method are in good agreement with both the exact solutions and the full DNS results, while the computational effort has been reduced by up to 72%. Linear stability analysis is presented in order to predict the stability of the method. Moreover, study of the interplay between the POD modes reveals the role of each dominant POD mode in dynamically representing the exact solution, and pinpoints the POD mode accounting for the instability in the approach.

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